Class WS note on continuous random variable
A random variable whose possible values form an interval is called a continuous random variable. For c.d.f. cumulative distribution function, FX ( x ) = P ( X x ) any continuous random variable Since the random variable is continuous the growth [0,1] is a continuous curve. There are no jumps. For a continuous random variable, the probability at any particular value is zero. P ( X = a ) = 0 for all a. We measure probability on intervals P ( a x b) = FX ( b ) FX ( a )
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P (a < x b) = FX ( b ) FX ( a ) P (a x < b) = FX ( b ) FX ( a ) P (a < x < b) = FX ( b ) FX ( a )
Since the probability at any particular value is zero the and < makes no difference to the probabilities
f X ( x ) is the probability density function. This is just the height of the function. The area under the curve equals 1. The height is not the probability at a point. We only can measure probability in intervals. The area under the curve between any two values is the probability that the event has occurred in that interval.
Example1: This is the f X ( x ) (p.d.f. )
a) Show the P (1.5 < X 4) on the graph b) What is P ( X = 1) c) Estimate P (1.5 < X 4)
Example 2: The following is a pdf: f(x) and its cdf: F(x) for a continuous random variable function a. Make an interval (a,b) on both graphs (on the horizontal) b. Show the P(a x b) on both graphs
f(x)
F(x)
x
x
Inclass Continuous Random variable
pg. 2
Uniform continuous random variable Example 3 The security guard will pass by my office between 12 -2pm. Each moment of time is equally likely. What is the probability that he will pass by my office between 12:15 and 1:12pm today? Need to find the amount of hours past noon: 12:15 is 0.25 of an hour and 1:12 is 1.2 hours
Graph the p.d.f
Write the function
f X ( x)
Graph the c.d.f.
Write the function FX ( x )
Evaluate the following b. f X (1.2 = ) a. f X (.25 ) =
c. P ( X = .25) =
d. P ( X = 1.2)
e.
=
f. FX (2) =
g. FX (.25) =
h. FX (1.2) =
Now we will answer the question:
What is the probability that he will pass by my office between 12:15 and 1:12pm today?
Find P (.25 X 1.2) There are two ways to calculate this
Inclass Continuous Random variable
pg. 3
Exponential random variable Example 4 Suppose that X is an exponential random variable, which has the following p.d.f. and c.d.f.
0 f X ( x ) = 1 7x 7 e
a) Sketch
x<0 x0
x<0 0 FX ( x ) = x 1 e 7 x 0
Sketch
f X ( x ) graph.
FX ( x ) graph.
Use your graphing calculator
Use your graphing calculator
b) Calculate
(5 decimal approximation)
c) Calculate
(5 decimal approximation)
Show the exact value of each probability and then give the 4 decimal place approximate is appropriate d) Sketch show and calculate this value
e) What is the probability that f) Calculate Only one method
X=3
g) What is the probability that the random variable is greater than 4
h) What is the mean of this distribution?
i) Is f X ( 4 ) = P ( X = 4) ? Explain
Inclass Continuous Random variable
pg. 4
Expected V...